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In the realm of probability, the **Probability Density Function (PDF)** of the Exponential Distribution tells us how likely it is for a particular event, say a specific wait time, to occur.
In this context, it represents the likelihood that a student uses a computer terminal for a particular amount of time.
The formula for the PDF in an exponential distribution is given by:\[f(x) = \lambda e^{-\lambda x}\]Where:

• \(f(x)\) is the probability density function.
• \(\lambda\) is the rate parameter, which is the reciprocal of the mean.
• \(x\) is the time or variable of interest.

The PDF is quite handy as it gives us a continuous curve that shows the likelihood of different outcomes.
In our example, with a mean of 36 minutes, the rate \(\lambda\) is \(\frac{1}{36}\).
Though the PDF itself doesn't give probabilities directly, it helps to generate the cumulative probabilities crucial for solving problems.

When we need to find out the overall probability of a variable being less than or equal to a certain value, we turn to the **Cumulative Distribution Function (CDF)**.
For the exponential distribution, this is useful for determining things like the probability of the wait time being 15 minutes or less.The CDF is expressed as:\[F(x) = 1 - e^{-\lambda x}\]Where:

• \(F(x)\) is the cumulative distribution function.
• \(e\) is the base of the natural logarithm.
• \(x\) is the variable or time of interest.

In essence, the CDF allows us to calculate the probability that the time until the event (such as waiting for the computer terminal) is less than or equal to \(x\).
For part (a) of the problem, using the mean of 36 minutes, we calculate \(P(X \leq 15)\) using the CDF to find the probability of waiting 15 minutes or less.
The formula simplifies calculations by helping you deduce probabilities over ranges of the distribution.

The **Exponential Probability Distribution** is paramount when dealing with supposedly random events that occur independently over time, like a student's usage time on a computer terminal.
The exponential distribution is defined by its mean and rate, \(\lambda\), which is \(\frac{1}{\text{mean}}\).
It's particularly well-suited for modeling survival times, duration of events, and wait times, due to its memoryless property.Some noteworthy features include:

• Its PDF and CDF are straightforward, as described before, making it ideal for exponential decay modeling.
• The mean and standard deviation of the exponential distribution are equal, which in our case is 36 minutes.
• Calculating probabilities, such as an event happening after a certain time, becomes easier with this distribution.

This memoryless property implies the distribution doesn't change even if we know prior events' occurrences.
It's spectacular for estimating how long one might wait before a new incident, like a student's wait time for a terminal.
Using this knowledge helps us solve parts of the exercise and understand real-world stochastic processes.